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Abstract

We both theoretically and experimentally examine the behavior of
the first- and the second-order rainbows produced by a normally
illuminated glass rod, which has a nearly elliptical cross section, as
it is rotated about its major axis. We decompose the measured
rainbow angle, taken as a function of the rod’s rotation angle, into a
Fourier series and find that the rod’s refractive index, average
ellipticity, and deviation from ellipticity are encoded primarily in
the m = 0, 2, 3 Fourier coefficients,
respectively. We determine these parameters for our glass rod and,
where possible, compare them with independent measurements. We find
that the average ellipticity of the rod agrees well with direct
measurements, but that the rod’s diameter inferred from the spacing of
the supernumeraries of the first-order rainbow is significantly larger
than that obtained by direct measurement. We also determine the
conditions under which the deviation of falling water droplets from an
oblate spheroidal shape permits the first few supernumeraries of the
second-order rainbow to be observed in a rain shower.

References

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Table 1

First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

E0

154.723

154.713

153.571

E2

-0.134

-1.346

-13.421

F2

-0.063

-0.636

-6.531

E4

2.3 × 10-4

0.021

2.104

F4

-4.1 × 10-5

-0.005

-0.477

E6

<10-6

-2.9 × 10-4

-0.296

F6

<10-6

3.2 × 10-4

0.332

E8

<10-6

<10-6

0.015

F8

<10-6

<10-6

-0.113

E2Mobius

-0.134

-1.340

-13.396

F2Mobius

-0.063

-0.633

-6.325

a The coefficients
E2Mobius and
F2Mobius are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ2D = 154.723°.

Table 2

First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

G0

262.121

262.159

266.370

G2

0.016

0.160

1.116

H2

-0.115

-1.162

-13.481

G4

2.4 × 10-4

0.021

1.980

H4

1.9 × 10-4

0.018

1.730

G6

<10-6

-4.2 × 10-4

-0.675

H6

<10-6

-9.4 × 10-4

-0.931

G8

<10-6

<10-6

0.009

H8

<10-6

<10-6

0.247

G2Mobius

0.016

0.160

1.599

H2Mobius

-0.115

-1.155

-11.546

a The coefficients
G2Mobius and
H2Mobius are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ3D = 262.121°.

Table 3

First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊ave =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

E0

154.580

154.580

154.580

154.579

154.577

E1

0.000

0.039

0.078

0.116

0.154

F1

0.000

0.011

0.022

0.034

0.044

E2

-5.019

-5.019

-5.021

-5.023

-5.025

F2

-2.378

-2.378

-2.378

-2.378

-2.379

E3

0.000

0.045

0.090

0.136

0.183

F3

0.000

0.034

0.069

0.104

0.139

E4

0.293

0.294

0.299

0.307

0.319

F4

-0.068

-0.066

-0.062

-0.055

-0.046

E5

0.000

-0.027

-0.054

-0.081

-0.108

F5

0.000

-0.046

-0.092

-0.137

-0.182

Table 4

First Six Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊ave =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

G0

262.657

262.638

262.619

262.565

262.489

G1

0.000

-0.011

-0.023

-0.034

-0.046

H1

0.000

0.034

0.068

0.102

0.134

G2

0.584

0.579

0.574

0.569

0.561

H2

-4.420

-4.408

-4.397

-4.381

-4.358

G3

0.000

0.575

1.150

1.724

2.294

H3

0.000

0.784

1.669

2.511

3.362

G4

0.301

0.310

0.320

0.353

0.395

H4

0.256

0.257

0.259

0.264

0.269

G5

0.000

0.120

0.240

0.359

0.479

H5

0.000

-0.168

-0.337

-0.505

-0.670

Table 5

First Six Fourier Coefficients in Degrees of the
Experimental First-Order Rainbow Deviation Angle and of
θ2R(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊ave = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

E0

153.990

154.579

E1

-0.071

0.101

F1

0.063

0.029

E2

-5.217

-5.022

F2

-1.909

-2.378

E3

0.131

0.118

F3

0.129

0.090

E4

-0.721

0.303

F4

0.514

-0.058

E5

-0.061

-0.071

F5

0.314

-0.119

Table 6

First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ3R(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊ave = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

G0

262.673

262.587

G1

-0.103

-0.030

H1

-0.283

0.092

G2

0.500

0.570

H2

-4.287

-4.389

G3

1.404

1.491

H3

2.093

2.177

G4

0.645

0.342

H4

-1.908

0.267

G5

-0.220

0.314

H5

0.380

-0.439

Tables (6)

Table 1

First Five Even Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

E0

154.723

154.713

153.571

E2

-0.134

-1.346

-13.421

F2

-0.063

-0.636

-6.531

E4

2.3 × 10-4

0.021

2.104

F4

-4.1 × 10-5

-0.005

-0.477

E6

<10-6

-2.9 × 10-4

-0.296

F6

<10-6

3.2 × 10-4

0.332

E8

<10-6

<10-6

0.015

F8

<10-6

<10-6

-0.113

E2Mobius

-0.134

-1.340

-13.396

F2Mobius

-0.063

-0.633

-6.325

a The coefficients
E2Mobius and
F2Mobius are obtained from Eq.
(18). The Descartes rainbow deviation angle is
θ2D = 154.723°.

Table 2

First Five Even Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(ξ) for an Elliptical
Cross-Sectional Cylinder with Refractive Index n =
1.474 and Ellipticities ∊ = -0.001, -0.01, and -0.1 as Defined in
Eq. (17)a

Coefficient

∊

-0.001

-0.01

-0.1

G0

262.121

262.159

266.370

G2

0.016

0.160

1.116

H2

-0.115

-1.162

-13.481

G4

2.4 × 10-4

0.021

1.980

H4

1.9 × 10-4

0.018

1.730

G6

<10-6

-4.2 × 10-4

-0.675

H6

<10-6

-9.4 × 10-4

-0.931

G8

<10-6

<10-6

0.009

H8

<10-6

<10-6

0.247

G2Mobius

0.016

0.160

1.599

H2Mobius

-0.115

-1.155

-11.546

a The coefficients
G2Mobius and
H2Mobius are obtained from Eq.
(19). The Descartes rainbow deviation angle is
θ3D = 262.121°.

Table 3

First Six Fourier Coefficients in Degrees of the
First-Order Rainbow Deviation Angle
θ2R(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊ave =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

E0

154.580

154.580

154.580

154.579

154.577

E1

0.000

0.039

0.078

0.116

0.154

F1

0.000

0.011

0.022

0.034

0.044

E2

-5.019

-5.019

-5.021

-5.023

-5.025

F2

-2.378

-2.378

-2.378

-2.378

-2.379

E3

0.000

0.045

0.090

0.136

0.183

F3

0.000

0.034

0.069

0.104

0.139

E4

0.293

0.294

0.299

0.307

0.319

F4

-0.068

-0.066

-0.062

-0.055

-0.046

E5

0.000

-0.027

-0.054

-0.081

-0.108

F5

0.000

-0.046

-0.092

-0.137

-0.182

Table 4

First Six Fourier Coefficients in Degrees of the
Second-Order Rainbow Deviation Angle
θ3R(ξ) for a
Two-Half-Ellipse Cross-Sectional Cylinder with Refractive Index
n = 1.474, Average Ellipticity ∊ave =
-0.037, and Various Values of the Ellipticity Difference Δ∊ as
Defined in Eq. (23)

Coefficient

Δ∊

0.00

0.01

0.02

0.03

0.04

G0

262.657

262.638

262.619

262.565

262.489

G1

0.000

-0.011

-0.023

-0.034

-0.046

H1

0.000

0.034

0.068

0.102

0.134

G2

0.584

0.579

0.574

0.569

0.561

H2

-4.420

-4.408

-4.397

-4.381

-4.358

G3

0.000

0.575

1.150

1.724

2.294

H3

0.000

0.784

1.669

2.511

3.362

G4

0.301

0.310

0.320

0.353

0.395

H4

0.256

0.257

0.259

0.264

0.269

G5

0.000

0.120

0.240

0.359

0.479

H5

0.000

-0.168

-0.337

-0.505

-0.670

Table 5

First Six Fourier Coefficients in Degrees of the
Experimental First-Order Rainbow Deviation Angle and of
θ2R(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊ave = -0.037, and
Ellipticity Difference Δ∊ = 0.026

Fourier Coefficient

Experiment

Theory

E0

153.990

154.579

E1

-0.071

0.101

F1

0.063

0.029

E2

-5.217

-5.022

F2

-1.909

-2.378

E3

0.131

0.118

F3

0.129

0.090

E4

-0.721

0.303

F4

0.514

-0.058

E5

-0.061

-0.071

F5

0.314

-0.119

Table 6

First Six Fourier Coefficients in Degrees of the
Experimental Second-Order Rainbow Angle and of
θ3R(ξ) for a Cylinder with
a Two-Half-Ellipse Cross Section, Refractive Index n =
1.474, Average Ellipticity ∊ave = -0.037, and
Ellipticity Difference Δ∊ = 0.026